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Originally called noughts and crosses (and still known as this in Britain and Australia), Xs and Os (in Ireland) and X and 0 (in India) is a pencil-and-paper game for two players, X and O, who take turns marking the spaces in a 3×3 grid. The player who succeeds in placing three respective marks in a horizontal, vertical, or diagonal row wins the game.

Despite its apparent simplicity, Tic-tac-toe requires detailed analysis to determine even some elementary combinatory facts, the most interesting of which are the number of possible games and the number of possible positions. A position is merely a state of the board, while a game usually refers to the way a terminal position is obtained.

Naive counting leads to 19,683 possible board layouts (39 since each of the nine spaces can be X, O or blank), and 362,880 (i.e. 9!) possible games (different sequences for placing the Xs and Os on the board). However, two matters much reduce these numbers:

The game ends when three-in-a-row is obtained.

The number of Xs is always either equal to or exactly 1 more than the number of Os (if X starts).

The complete analysis is further complicated by the definitions used when setting the conditions, like board symmetries.

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